Yield surfaces in prestrained aluminum and copper
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&HE a n a l y s i s of s t r a i n - h a r d e n i n g solids s u b j e c t e d to m u l t i a x i a l s t a t e s of s t r e s s r e q u i r e s knowledge of the following r e l a t i o n s : 1) An i n i t i a l yield condition, s p e c ifying the state of s t r e s s at which p l a s t i c d e f o r m a t i o n b e g i n s ; 2) a flow r u l e , specifying the r e l a t i o n s h i p b e tween s t r e s s and the i n c r e m e n t s of p l a s t i c s t r a i n d u r ing p l a s t i c flow; 3) a h a r d e n i n g r u l e , specifying the change in the i n i t i a l yield condition with continued p l a s t i c d e f o r m a t i o n . The l a s t of t h e s e r e l a t i o n s n e e d s the m o s t e x p e r i m e n t a l study b e c a u s e data on the shape of y i e l d s u r f a c e s of r e a l m a t e r i a l s a s a function of various strain histories are timited. If the effects of t i m e and t e m p e r a t u r e and the i n fluence of the h y d r o s t a t i c s t r e s s component on p l a s t i c flow can be neglected, the yield l o c u s of a m a t e r i a l can be r e p r e s e n t e d by a function of the i n v a r i a n t s of the s t r e s s d e v i a t o r t e n s o r . A c c o r d i n g to D r u c k e r ' s s t a b i l i t y p o s t u l a t e , t such a yield locus m u s t be ext e r n a l l y convex, and the r e s u l t i n g p l a s t i c s t r a i n i n c r e m e n t d u r i n g p l a s t i c flow m u s t be n o r m a l to the y i e l d locus. The convexity r u l e does not p r o h i b i t the e x i s t e n c e of c o r n e r s in the yield surface. However, the d i r e c t i o n of the p l a s t i c s t r a i n i n c r e m e n t at such c o r n e r s would be i n d e t e r m i n a t e . The two m o s t c o m m o n l y u s e d i n i t i a l yield conditions a r e the yon M i s e s o r J2 y i e l d c r i t e r i o n , which s t a t e s that yielding o c c u r s when J2, the second i n v a r i a n t of the s t r e s s d e v i a t o r t e n s o r , r e a c h e s a c r i t i c a l v a l u e , and the T r e s c a or m a x i m u m s h e a r s t r e s s c r i t e r i o n , which s t a t e s that yielding o c c u r s when the m a x i m u m s h e a r s t r e s s r e a c h e s a c r i t i c a l value. T h e r e is l i t t l e d i s a g r e e m e n t that most of the data of s t r a i n - h a r d e n i n g m a t e r i a l s fall between the two t h e o r e t i c a l p r e d i c t i o n s but a r e g e n e r a l l y c l o s e r to the yon M i s e s c r i t e r i o n . S. S. HECKER, formerly Postdoctoral Appointee at the Los Alamos Scientific Laboratory, Los Alamos, N. Mex., is now Senior Research Metallurgist, Physics Department, General Motors Research Laboratories, Warren,Mich. Manuscript submitted December 22, 1970. METALLURGICALTRANSACTIONS
T h e r e is, h o w e v e r , c o n s i d e r a b l e d i s a g r e e m e n t about the shape of s u b s e q u e n t yield s u r f a c e s a s a function of p l a s t i c s t r a i n h i s t o r y . N u m e r o u s a t t e m p t s have b e e n m a d e to e s t a b
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